going bananas

"I've got ever such a big bag of bananas here (in my arms)
can you see them?
no?
that's because it's a pretend story...

and there's a heap of bananas over here (gesture)
can you see it?
no?
that's because it's a potential pile at the moment....










you are only allowed to do two things to this pile to change it:

A: add 3 bananas
B: take away 7 bananas

A: add 3 bananas

B is the same as adding 7 anti-bananas

one anti-banana
cancels out a normal banana

7 anti-bananas
cancel out7 normal bananas

to begin more simply,
it is easier to use A : add 2
and B : take away 3
NRICH do this in their 'Strange Bank Account' resources
with a fine introductory video clip

back to these rules of either adding 3's or subtracting 7's,
you can do either of these things:

• A (add 3) or 
• B (take away 7) 

as many times as you like

starting from a pile with nothing in it
how can you end up with just 1 banana in the pile?
5A + 3B
or?
various options,
e.g. 12A + 5B

what other numbers of bananas can be made?
1 = 5A + 2B
2 = 3A + B
3 = A
etc.

students continue, trying to make all positive integers, for a while
they might be able to detect patterns

"hang on a minute, if 1 = 5A + 2B how could you get 2? [10A + 4B]
but we said that 2 = 3A + B
so that means there are different ways of making some of these numbers in the pile..."

"how did anyone make 5?"
  4A +   B = 5
11A + 4B = 5
18A + 7B = 5
how does this pattern continue?
why does it work with these numbers?
what happens if we put the pattern into reverse? what is the expression before 4A + B?
why does this work?
what is the next one before this? etc.

let's try to establish (prove) that you can make all of the positive integers
[they can either be derived from 1 or since 1, 2 and 3 can be made so can 4 (1 + 3) etc.]

how can we make -1? [ an anti-banana pile]?
-2? etc.

what is the connection between a positive number pile and its opposite?
 1 = 5A + 2B
-1 = 2A +  B

-2 = 4A +2B and 2 = 3A + B

-3 = 6A + 3B and 3 = A

why does the connection work?

what could A - B mean?
but this is the same as 8A + 2B?
why?

extend to other pairs of operations for A and B...